To compress the graph of $f(x) = x^2$ horizontally by a factor of $\frac{1}{3}$, we apply a horizontal scaling transformation.

Key Idea:

For a horizontal compression or stretch, the equation becomes: $g(x) = f\left(\frac{x}{k}\right)$ where $k > 1$ compresses the graph horizontally, and $k < 1$ stretches it.

Here, the horizontal compression factor is $\frac{1}{3}$, so we set $k = 3$ because: $x \rightarrow 3x \quad \text{compresses the graph horizontally}.$

New Equation:

The new function $g(x)$ is: $g(x) = f(3x)$ Substitute $f(x) = x^2$: $g(x) = (3x)^2$ Simplify: $g(x) = 9x^2$

Final Answer:

The equation for $g(x)$ is: $g(x) = 9x^2$

Graph Effect:

• The parabola becomes narrower because of the horizontal compression.
• Every point on the graph moves closer to the y-axis.

Would you like a graphical comparison or further explanation of transformations?

Related Questions:

1. How would you stretch the graph horizontally by a factor of 2?
2. What would happen if you reflected the graph across the y-axis?
3. How do you vertically stretch $f(x) = x^2$ by a factor of 4?
4. What is the equation for a horizontal shift of $f(x)$ 5 units to the right?
5. How do combined transformations (compression and vertical shifts) affect the graph?

Tip:

Remember, horizontal transformations involve scaling the input (x), while vertical transformations scale the output.